Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 1 - First-Order Differential Equations - 1.2 Basic Ideas and Terminology - Problems - Page 22: 42



Work Step by Step

Integrate the function to turn $y''$ into $y'$. $$y''=e^{-x}$$ $$y'=-e^{-x}+C_1$$ Integrate once again to turn $y'$ into $y$. $$y=e^{-x}+C_1x+C_2$$ Solve for the initial values. $$y(0)=e^{-0}+C_1(0)+C_2=1+C_2$$ $$1+C_2=y(0)=1$$ $$C_2=0$$ $$y(1)=e^{-1}+C_1+C_2$$ $$0=e^{-1}+C_1+C_2$$ $$C_1=-e^{-1}$$ Substitute these constant values to get $y=e^{-x}-e^{-1}x$.
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